Define g{{\left({x},{y}\right)}}={x}^{2}+{y}^{2}-{4}{x}{y}+{3}{y}+{2}

cistG 2021-09-10 Answered

Define \(\displaystyle g{{\left({x},{y}\right)}}={x}^{2}+{y}^{2}-{4}{x}{y}+{3}{y}+{2}\)

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Expert Answer

Tuthornt
Answered 2021-09-11 Author has 13696 answers

Possible derivation:
\(\displaystyle\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}+{y}^{2}-{4}{x}{y}+{3}{y}+{2}\right)}\)
Differentiate the sum term by term and factor out constants:
\(\displaystyle=\frac{d}{{\left.{d}{x}\right.}}{\left({2}\right)}-{4}{y}{\left(\frac{d}{{\left.{d}{x}\right.}}{\left({x}\right)}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({3}{y}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}\)
The derivative of 2 is zero:
\(\displaystyle=-{4}{y}{\left(\frac{d}{{\left.{d}{x}\right.}}{\left({x}\right)}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({3}{y}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}+{0}\)
Simplify the expression:
\(\displaystyle=-{4}{y}{\left(\frac{d}{{\left.{d}{x}\right.}}{\left({x}\right)}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({3}{y}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}\)
The derivative of x is 1:
\(\displaystyle=\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({3}{y}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}-{1}{4}{y}\)
Use the power rule, \(\displaystyle\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{n}\right)}={n}{x}^{{{n}-{1}}}\), where n = 2.
\(\displaystyle\frac{d}{{\left.{d}{x}\right.}}{\left({x}^{2}\right)}={2}{x}:\)
\(\displaystyle=-{4}{y}+\frac{d}{{\left.{d}{x}\right.}}{\left({3}{y}\right)}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}+{2}{x}\)
The derivative of 3 y is zero:
\(\displaystyle={2}{x}-{4}{y}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}+{0}\)
Simplify the expression:
\(\displaystyle={2}{x}-{4}{y}+\frac{d}{{\left.{d}{x}\right.}}{\left({y}^{2}\right)}\)
The derivative of \(\displaystyle{y}^{{{2}}}\) is zero:
\(= 2 x - 4 y + 0\)
Simplify the expression:
\(= 2 x - 4 y\)
Simplify the expression:
Answer:
\(= 2 (x - 2 y)\)

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